Question

a farmer has 3600 feet of fence to enclose a rectangular area. what dimensions for the rectangle result in the maximum area enclosed by the fence? the rectangle with the maximum area enclosed by the fence has a length of (square) ft and a width of (square) ft. (simplify your answers.)

Explanation:

Step1: Let the length be $l$ and width be $w$.

The perimeter of a rectangle is $P = 2l + 2w$. Given $P=3600$, so $2l + 2w=3600$, which simplifies to $l + w = 1800$, and $l=1800 - w$.

Step2: Express the area formula.

The area of a rectangle $A=l\times w$. Substitute $l = 1800 - w$ into the area formula, we get $A=(1800 - w)w=1800w - w^{2}$.

Step3: Find the maximum of the area function.

Since $A(w)=1800w - w^{2}$ is a quadratic - function of the form $y = ax^{2}+bx + c$ with $a=-1$, $b = 1800$, $c = 0$. The vertex of a quadratic function $y = ax^{2}+bx + c$ has its $x$ - coordinate (in our case $w$ - coordinate) at $w=-\frac{b}{2a}$. So $w=-\frac{1800}{2\times(-1)} = 900$.

Step4: Find the length.

Substitute $w = 900$ into $l=1800 - w$, we get $l=1800 - 900=900$.

Step5: Calculate the maximum area.

$A=l\times w=900\times900 = 810000$ square - feet.

Answer:

The length is 900 ft and the width is 900 ft, and the maximum area is 810000 square feet.